NoIndependentVariable finds the solution of an ODE which doesn't contain the independent variable other than via the dependent variable and its derivatives.

The second argument, y(x), representing the variable for the existing ODE, is optional. It must be given however, if the dependent and independent variables cannot be determined from the ODE.

The third argument, u(t), representing the variable for the reduced ODE, is optional. If it is not given, new independent and dependent variables will be chosen which do not conflict with the existing variables.

The default output is a sequence consisting of the reduced ODE in terms of the new variables, followed by the transformation used to recover the original ODE from the reduced ODE.

If an extra option solve or solve=true is also given, an attempt is made to solve the reduced ODE and return the general solution to the original ODE. If successful, the general solution of the original ODE will be returned.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\right)\:$

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{ReduceOrder}\right]\right)\:$

$\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+\frac{{\mathrm{diff}\left(y\left(x\right)\,x\right)}^2}{y\left(x\right)}=0$

$\mathrm{ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\frac{{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}{y\left(x\right)}=0$

$\mathrm{reduced\_ode},\mathrm{tr}≔\mathrm{NoIndependentVariable}\left(\mathrm{ode}\right)$

$\mathrm{reduced\_ode},\mathrm{tr}≔u\left(t\right)\left(\frac{\ⅆ}{\ⅆt}u\left(t\right)\right)+\frac{{u\left(t\right)}^2}{t}=0,\left\{t=y\left(x\right)\,u\left(t\right)=\frac{\ⅆ}{\ⅆx}y\left(x\right)\right\}$

$\mathrm{reduced\_sol}≔\mathrm{Solve}\left(\mathrm{reduced\_ode}\,u\left(t\right)\right)$

$\mathrm{reduced\_sol}≔u\left(t\right)=\frac{{\ⅇ}^{\mathrm{\_C1}}}{t}$

$\mathrm{new\_ode}≔\mathrm{eval}\left(\mathrm{reduced\_sol}\,\mathrm{tr}\right)$

$\mathrm{new\_ode}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=\frac{{\ⅇ}^{\mathrm{\_C1}}}{y\left(x\right)}$

$\mathrm{Solve}\left(\mathrm{new\_ode}\,y\left(x\right)\right)$

$\left\{y\left(x\right)=\sqrt{2{\ⅇ}^{\mathrm{c\_\_1}}x+2\mathrm{\_C2}}\,y\left(x\right)=-\sqrt{2{\ⅇ}^{\mathrm{c\_\_1}}x+2\mathrm{\_C2}}\right\}$

$\mathrm{NoIndependentVariable}\left(\mathrm{ode}\,'\mathrm{solve}'\right)$

$\left\{y\left(x\right)=\sqrt{2{\ⅇ}^{\mathrm{c\_\_1}}x+2\mathrm{\_C2}}\,y\left(x\right)=-\sqrt{2{\ⅇ}^{\mathrm{c\_\_1}}x+2\mathrm{\_C2}}\right\}$

$\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+y\left(x\right)\mathrm{diff}\left(y\left(x\right)\,x\right)=0$

$\mathrm{ode2}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)+\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)y\left(x\right)=0$

$\mathrm{reduced\_ode2},\mathrm{tr2}≔\mathrm{NoIndependentVariable}\left(\mathrm{ode2}\,y\left(x\right)\,v\left(s\right)\right)$

$\mathrm{reduced\_ode2},\mathrm{tr2}≔v\left(s\right)\left(\frac{\ⅆ}{\ⅆs}v\left(s\right)\right)+v\left(s\right)s=0,\left\{s=y\left(x\right)\,v\left(s\right)=\frac{\ⅆ}{\ⅆx}y\left(x\right)\right\}$

$\mathrm{reduced\_sol2}≔\mathrm{Solve}\left(\mathrm{reduced\_ode2}\,v\left(s\right)\right)$

$\mathrm{reduced\_sol2}≔v\left(s\right)=-\frac{s^2}{2}+\mathrm{c\_\_1}$

$\mathrm{new\_ode2}≔\mathrm{eval}\left(\mathrm{reduced\_sol2}\,\mathrm{tr2}\right)$

$\mathrm{new\_ode2}≔\frac{\ⅆ}{\ⅆx}y\left(x\right)=-\frac{{y\left(x\right)}^2}{2}+\mathrm{c\_\_1}$

$\mathrm{Solve}\left(\mathrm{new\_ode2}\,y\left(x\right)\right)$

$y\left(x\right)=\tanh \left(\frac{\sqrt{\mathrm{c\_\_1}}\left(x+\mathrm{\_C2}\right)\sqrt{2}}{2}\right)\sqrt{\mathrm{c\_\_1}}\sqrt{2}$

$\mathrm{NoIndependentVariable}\left(\mathrm{ode2}\,'\mathrm{solve}'\right)$

$y\left(x\right)=\tanh \left(\frac{\sqrt{\mathrm{c\_\_1}}\left(x+\mathrm{\_C2}\right)\sqrt{2}}{2}\right)\sqrt{\mathrm{c\_\_1}}\sqrt{2}$

The Student[ODEs][ReduceOrder][NoIndependentVariable] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.